Beamforming networks, such as a phased-array beamforming network, may require the calibration of the elements of the antenna array. This may involve the removal of any unknown and undesired amplitude and phase offsets between the antenna array elements. For example, in the return link of a Ground-Based Beamforming (GBBF) network of a satellite communication system, calibration may be achieved by having several geographically separated Beacon Transmitters (BT) transmitting calibration beacon signals on each antenna element path and received by a single Measurement Node (MN). Each BT may have good visibility only to a subset of antenna elements. Furthermore each BT may introduce an unknown nuisance perturbation or an offset in the transmitted signal. The objective may be to calibrate all elements of the antenna array based on measurements made at the MN of signals arriving from geographically separated BTs.
Similarly, in the forward link of a Ground-Based Beam Forming (GBBF) network of a satellite communication system, the array calibration may be achieved by a Beacon Transmitter (BT) transmitting, on each antenna element path, a calibration beacon signal which may be received and measured at several geographically separated Measurement Nodes (MN). Each MN may have good visibility only to a subset of antenna elements. Further, measurement at each MN may be affected by an unknown offset plus varying amounts of noise. The objective may be to calibrate all elements of the antenna array based on measurements made locally at each MN.
Other systems in which data from many separate sensors must be coherently combined and compared in order to determine the required system parameters may use parameter estimation. Such systems include, for example, surveying systems, in which many separate measurements may be combined to make a complete measurement, and various wireless sensor networks. The outputs of each MN may be affected by some unknown nuisance parameter, such as, for example, common measurement instrument errors (such as altitude errors that cannot be measured in surveying systems). Other applications for parameter estimation include channel estimation in Distributed MIMO (D-MIMO) architectures, and blind system identification using multiple MNs in scenarios where parameters representing the system under test may be reliably identified using multiple MNs, and where each individual MN may introduce uncertainty in its measurements. This may include estimating possibly complex-valued voltages in an electric circuit using voltage-meters that are not well calibrated.
Parameter estimation may be accomplished using selective daisy chaining, which may include combining the measurement sets. Selective daisy chaining arrives at the estimation of a parameter set, denoted by a vector, by tracing out several of the least-noisy paths in a bipartite network graph of parameters nodes and BT nodes on the return link (or MNs on the forward link), connecting the parameters nodes and the BTs (or MNs). Selective daisy chaining may lack accuracy and robustness to instrument failures, since it may not use all of the available information in all the measurement sets, and may throw away the information not belonging to the selected paths.
An extension of selective daisy chaining include the maximal daisy chain approach, which may be equivalent to a maximum ratio combining scheme, and path search techniques on the graph problems, in which all the possible paths between each pair of the parameter nodes, through all the different BTs (or MNs), may be traced out and combined using a reliability metric of all the measurements encountered along each path. Selective daisy chaining may be computationally complex, and may be an NP-hard problem that becomes unwieldy as the number of the parameters and BTs (or MNs) increases.
A linear least-squares approach to combining the measurement sets (that are collected using either the BTs on the return link or the MNs on the forward link) may also be used for parameter estimation. However, linear least-squares may have a phase ambiguity problem in the context of the parameter estimation problem in the present invention, and therefore may not be used for the estimation of complex-valued channel coefficients. The linear least-squares approach may be feasible only for the linear model of the measurements.